I remember waking up one morning to hear the score from the west coast games. Heard the Dodgers won 1-0. The winning pitcher was Don Drysdale and the guy who hit the winning HR was Don Drysdale.
Nope.
Don Drysdale pitched a lot of shutouts, (49), and hit a lot of home runs, for a pitcher, (29), but...
He didn't hit any home runs after the 1966 season. In 1965 and and 1966, Drysdale DID hit home runs in games in which he threw a shutout, but neither was a 1-0 game, and both games were played in the afternoon, so those games would have been complete before you went to bed that night, probably. Unless this happened in 1961 or earlier, this is a false memory.
On
June 23, 1971, Rick Wise of the Philadelphia Phillies pitched a no-hitter, AND hit TWO home runs, with 3 RBIs in a 4-0 win over the Reds, which is a more impressive feat. Wise had a decent but unremarkable career after 1971, but he's best remembered for having been traded to the St Louis Cardinals before the 1972 season, even up for Steve Carlton.
This was NOT the only time the Cardinals traded away a pitcher who would win the Cy Young Award in his first year with his new team. In one deal, they traded away the Cy Young Award winners for the subsequent two years, but that's a story for the New Sports Trivia thread.